Let R = k[[x_0,...,x_d]]/(f), where k is a field and f is a non-zero non-unit
of the formal power series ring k[[x_0,...,x_d]]. We investigate the question
of which rings of this form have bounded Cohen--Macaulay type, that is, have a
bound on the multiplicities of the indecomposable maximal Cohen--Macaulay
modules. As with finite Cohen--Macaulay type, if the characteristic is
different from two, the question reduces to the one-dimensional case: The ring
R has bounded Cohen--Macaulay type if and only if R is isomorphic to
k[[x_0,...,x_d]]/(g+x_2^2+...+x_d^2), where g is an element of k[[x_0,x_1]] and
k[[x_0,x_1]]/(g) has bounded Cohen--Macaulay type. We determine which rings of
the form k[[x_0,x_1]]/(g) have bounded Cohen--Macaulay type.Comment: 16 pages, referee's suggestions and correction
